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PostSubject: Material conditional   Sun Nov 21, 2010 9:10 am

The material conditional, also known as material implication, is a binary truth function → such that the compound sentence p→q (typically read "if p then q") is logically equivalent to the negative compound: not(p and not q). A material conditional compound itself is often simply called a conditional. By definition of "→", the compound p→q is false if and only if both p is true and q is false. That is to say that p→q is true if and only if either p is false or q is true (or both). Thus → is a function from pairs of truth values of the components p, q to truth values of the compound p→q, whose truth value is entirely a function of the truth values of the components. Thus p→q is said to be truth-functional. p→q is logically equivalent also to ¬p∨q (either not p, or q (or both)), and to ¬q → ¬p (if not q then not p), but not to ¬p → ¬q. For convenience, p→q is typically read "If p, then q", "p only if q", or "q if p". Saying "It is false that if p then q" does not always sound logically equivalent in everyday English to saying "both p and not q" but, when used in logic, it is taken as logically equivalent. (Other senses of English "if...then..." require other logical forms.)

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PostSubject: Re: Material conditional   Tue Dec 21, 2010 5:19 am

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